Having the Queen pretending to be the leader of the Zygons, somehow knowing exactly what their plan was, and pulling it off well enough to fool the other Zygons.
As cool as it was to have all thirteen Doctors join forces to save Gallifrey, some have questioned the credibility of the scene. How in the hell did they manage to summon all thirteen Doctors at once, and got them to agree to come over? Did Eleven travel back and visit each of them himself, or did the Moment bring them altogether? And what about the major repercussions of crossing his own timeline? Given that three Doctors was catastrophic enough already, having all thirteen together alone would destroy reality.
Billie Piper as The Moment's body. Some don't mind it, others... still have issues over Rose. Although, there were some who disliked Rose as a character but thought Piper as The Moment was fantastic. And, technically speaking, The Moment wasn't actually Rose, but the Bad Wolf entity (which possessed Rose). Then there were people who were unhappy Rose wasn't in the special, feeling they should have had her and Doctor 10.5.
The idea of the War Doctor. Some say John Hurt was fantastic in the role, others believe that the character could've been filled with either Paul McGann (as number 8) or Christopher Eccleston (as number 9). While both Doctors do appear in the story in brief but crucial roles, it's only through stock footage, and the former already made an appearance in a lead-up mini-sode.
Moreover, Eccleston declined to return for the special. Thus, we never get to dwell on his Doctor finding out he never really destroyed his own planet and people, though he's going to forget anyway. It would have been fitting as he's the first Doctor shown to bear that guilt ever since the show was revived.
The Ninth Doctor was also implied to have recently regenerated way back in Series 1 (surprised to see his big ears), so it was naturally assumed the Eighth Doctor ended the Time War. Regardless, Moffat was having trouble picturing the Eighth Doctor fighting in a war because of what his character was like, even when Eight started turning darker. Which led to a prequel mini-episode with a regeneration scene for his Doctor to explain away the War Doctor's existence, so it's still a win in the end for fans who wanted to see McGann again, along with making his Big Finish Doctor Who adventures canon. And Hurt's Doctor even goes out on a line that links back to Nine's ears comment.
Clara suddenly being able to close the TARDIS' doors with a snap of her fingers), doing cool stunts on her motorcycle and managing to talk the Doctor down from burning Gallifrey again. Audience reactions ranged from "Wow!" to "What the hell?!" She's becoming very divisive in the fanbase. Part of the issue is that the finger snapping to close the door was a big deal for the Doctor to do, so for Clara to just casually do it is a bit off-putting. That and it marks the fact that the TARDIS has apparently done a 180 on its opinion of her (it had previously blatantly hated her) because the paradoxes surrounding Clara have been resolved! It's probably also because Clara (i.e. the Gallifreyian version of her) introduced the Doctor to his TARDIS. Though that brings up it's own issues since the TARDIS explicitly stated it chose the Doctor.
Some fans of the classic Who felt the special was more of an anniversary special on the new era rather than a celebration of all of Who history outside a few cameos and the fact that the event that separates old from new who is the focus of the plot. Fans who joined in during the reboot pointed out that the Big Finish Audio Drama, "Night of the Doctor" and The Five(ish) Doctors Reboot were all more focused on Classic Who. On the other hand, the cameo of Tom Baker, plus the quips of the War Doctor at the behaviors of Ten and Eleven were great big nods at the classic roots.
The reveal that the Doctors saved Gallifrey instead of destroying it. Some thought it was perfectly in character for the Doctor to Take a Third Option instead of needlessly killing millions of innocent lives. Species and characters who seem to be Killed Off for Real only to be Not Quite Dead or Back from the Dead have always been a staple of the series since day one. After all if the Daleks, Cybermen, Davros, the Master (and the planet Skaro) can keep coming back after being destroyed time and time again, why not the Time Lords and Gallifrey? Still diehard Russell T. Davies fans were furious at what they saw as Moffat needlessly tampering with an essential part of the Doctor's backstory.
Narm: The repeated shots of the children of Gallifrey to some, who felt it bordered on glurge.
Heartwarming in Hindsight: The Face of Boe's message in "Gridlock" that, "You are not alone," can take on a much more positive meaning with this episode, since it can be interpreted now as a message that Gallifrey is safe.
Hollywood Homely: Osgood. The Zygon version notes how jealous Osgood is of her prettier sister, but the actress playing her is a pretty actress in nerd garb.
Hype Backlash: The Time War, built up as a war the likes of which there has never been nor will ever be. Unfathomable atrocities were committed by both sides as the lines between good and evil were all but eradicated. The Battle for Gallifrey was where it all was turned Up to Eleven as the Doctor ultimately decided that committing genocide on both sides was the only way to stop the entire universe from succumbing to a fate worse than death. What we got was…a pretty standard shootout between Daleks and Time Lords (not to mention one that lacks time as an element of warfare).
Russell T. Davies went on record saying that when he was in charge he avoided showing anything from the Time War because it would inevitably wind up creating this.